Advantages

• Best for studies involving populations of rare and clustered individuals
• Compared to designs such as simple random sampling:
  • Increased sampling efficiency
  • Reduced variance

Disadvantage

• Final sample size is unknown at the start of sampling

• same procedure as ACS, except that we restrict the number of neighboring units to \(q\) (in our example, \(q = 12\))

• The unbiased (for ACS) Horvitz-Thompson estimator of the population mean of variable \(x\) is \(\bar{x}_{HT} = \frac{1}{N}\sum_{i=1}^{v} (x_i J_i)/\pi_i\)

  • \(N\) = population size, \(n_1\) = primary sample size
  • \(v\) = the number of unique units in the sample
  • \(J_i\) is an indicator variable, equalling 1 if the \(i\)^th unit in the sample satisfies the condition or was included in the primary sample and 0 otherwise
  • inclusion probability: \(\pi_i = 1 - \binom{N - m_i}{n_1} \Big/ \binom{N}{n_1}\), where \(m_i\) is the number of units in the network \(\psi_i\)

• Objective: Explore properties of the RACS design by bootstrapping populations created with a dataset of two cactus species and their associated insects.
• randomly sampled the survey data to generate 6 populations

Horvitz-Thompson estimators will be

1. biased for the variable on which the RACS criterion is based
   • i.e., cactus presence
2. unbiased for variables that
   • are not used to generate the RACS criterion
   • are uncorrelated with the variable on which the RACS criterion is based
   • i.e., insect presence, evidence of insect damage

• We used cactus presence as our criterion to initiate cluster sampling

• We ran 5000 simulations of the RACS and ACS designs per

  • population
  • primary sample size (20, 30, 40, 75, and 100)

• For each simulation we estimated:

  • O. stricta, O. pusilla, and cactus occupancy
  • insect occupancy given cactus presence

• relative bias of the mean:

\(B_{\bar{x}_{HT,i}} = (\bar{x}_{HT,i} - \mu)/\mu\)

• \(\mu\) is the true population mean

• \(\bar{x}_{HT,i}\) Horvitz-Thompson mean of variable \(x\) from the \(i\)^th simulation

• mean percent relative bias for the \(S\) simulations

\(\text{mean}(B_{\bar{x}_{HT}}) = 100 \times S^{-1} \sum_{i=1}^{S} B_{\bar{x}_{HT,i}}\)

• with RACS, we underestimate occupancy (negative percent relative bias)
• this underestimate increases with cactus occupancy

• Variability decreases with increased primary sample size and occupancy

• bias decreases with increasing primary sample size
• noise suggests that something other than just occupancy (e.g., spatial structure) is influencing the bias

• Compared to ACS, RACS generates smaller final sample sizes, particularly for populations with relatively high occupancy
• confirmed that estimate of cactus occupancy is biased
  • may need to fix calculation so that bias does not increase with \(n_1\)
• with larger sample sizes, our estimates of moth occupancy is less biased

• correct bias by correcting inclusion/intersection probabilities
• ways to do this:
  • analytical - come up with a formula based on average behavior
  • use nonparametric technique such as bootstrapping
    • keep track of how many times a unit is sampled we can estimate the probabilities

• we have also looked at bias for:
  • variance
  • insect damage occupancy
  • max. plant height
  • percentage of plot covered by cacti
• we have created a R package, "ACSampling," that we will release as a public Github repository

• Estimate bias of the variance
• Run more simulations
• Test bias reduction techniques
• compare efficiency of RACS, ACS, and SRSWOR
  • primary sample size = 40 + RACS = 40 vs. SRSWOR = 80?